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Dense Sets, Nowhere Dense Sets and an Ideal in Generalized Closure Spaces

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Dense Sets, Nowhere Dense Sets and an Ideal in Generalized Closure Spaces MATEMATIQKI VESNIK UDK 515.122 59 (2007), 181–188 originalni nauqni rad research paper DENSE SETS, NOWHERE DENSE SETS AND AN IDEAL IN GENERALIZED CLOSURE SPACES Chandan Chattopadhyay Abstract. In this paper, concepts of various forms of dense sets and nowhere dense sets in generalized closure spaces have been introduced. The interrelationship among the various notions have been studied in detail. Also, the existence of an ideal in generalized closure spaces has been settled. 1. Introduction Structure of closure spaces is more general than that of topological spaces. Hammer studied closure spaces extensively in [8,9], and a recent study on these spaces can be found in Gnilka [5,6], Stadler [14,15], Harris [10], Habil and Elzena- ti [7]. Although the applications of general topology is not available so much in digital topology, image analysis and pattern recognition; the theory of generalized closure spaces has been found very important and useful in the study of image analysis [3,13]. In [14,15], Stadler studied separation axioms on generalized closure spaces. The following definition of a generalized closure space can be found in [7] and [15]. Let X be a set. }(X) be its power set and cl : }(X) ¡! }(X) be any arbitrary set-valued set function, called a closure function. We call clA, A ½ X, the closure of A and we call the pair (X; cl) a generalized closure space. The closure function in a generalized closure space (X; cl) is called: (a) grounded if cl(;) = ;, (b) isotonic if A ½ B ) clA ½ clB, (c) expanding if A ½ clA for all A ½ X, (d) sub-additive if cl(A [ B) ½ clA [ clB, (e) idempotent if cl(clA) = clA, S S (f) additive if ¸2Λ cl(A¸) = cl( ¸2Λ(A¸)). AMS Subject Classification: 54A05. Keywords and phrases: Generalized closure spaces, isotonic spaces, dense sets, nowhere dense sets, ideals. 181 182 Ch. Chattopadhyay A generalized closure space will be called: (i) grounded, if its closure function is grounded, (ii) isotonic, if its closure function is grounded and isotonic, (iii) expanding, if its closure function is expanding, (iv) idempotent, if its closure function is idempotent. A isotonic expanding space (X; cl) is called a neighbourhood space. An idem- potent neighbourhood space is called a closure space. A sub-additive closure space is a topological space. The interior function int : }(X) ¡! }(X) is defined by intA = X ¡cl(X ¡A). It follows that clA = X ¡ (int(X ¡ A)) for all A ½ X. A set A 2 }(X) is called closed in the generalized closure space (X; cl) if clA = A holds. A is called open if X ¡ A is closed or if A = intA. We note that in a topological space(X; ¿), a dense subset D of X satisfies the following properties: (1D) for every nonempty open set V in ¿, V \ D 6= ;; (2D) clD = X; (3D) for any superset B of D, B is dense in ¿; (4D) int(X ¡ D) = ;. Now the question naturally arises whether the concept of a dense set in a generalized closure space can be given which will, under certain restrictions on the generalized closure space, satisfy the above four parallel properties. Again we note that in a topological space, a nowhere dense set A satisfies the following properties: (1N) for every nonempty open set V , there exists a nonempty open set W ½ V such that W \ A = ;, (2N) int clA = ;, (3N) for every subset B of A, B is nowhere dense, (4N) cl int(X ¡ A) = X. Now the problem is to define nowhere dense sets in a generalized closure space which, under certain restrictions on the generalized closure space, will satisfy the above four parallel properties. Ideals play an important role in topological spaces. Ideals are used as an indispensable tool in constructing new topologies from old [11], in the study of I-resolvability [2], I-compactification and local I-compactness [12], in the study of I-continuity [1], in the study of Baire spaces and Volterra spaces [4] etc. Various collections of ideals have been investigated and used in the study of the topics just mentioned. One of the most important and useful example of an ideal in a topological space (X; ¿) is the collection of nowhere dense sets in (X; ¿). Now the following question arises: Whether there exists an ideal in non-topological generalized closure spaces? Dense sets, nowhere dense sets and an ideal in generalized closure spaces 183 A partial answer to this question has been given in this paper. Henceforth, a generalized closure space will be written as a gc-space. 2. Dense sets in gc-spaces It is natural to define a dense set in a gc-space as follows: Definition 2.1. A nonempty subset D of X is called gc-dense in a gc-space (X; cl) if V \ D 6= ; for every nonempty open set V in (X; cl). Then it follows that if D is gc-dense then for any superset B of D, B is gc-dense in (X; cl). But property (2D) may not hold. Example 2:1. Let X = fa; b; cg. Define cl : }(X) ¡! }(X) by cl; = ;, clX = X, clfag = X, clfbg = fbg, clfcg = fcg, clfa; bg = fa; bg, clfa; cg = fa; bg, clfb; cg = fb; cg. Note that nonempty open sets are X, fa; cg, fa; bg, fcg, fag. Now if A = fa; cg then V \ A 6= ; for every nonempty open set V in (X; cl), but clA = fa; bg 6= X. The above example allows us to define a new concept of dense set in a gc-space. Definition 2.2. A nonempty subset D of X will be called sgc-dense in a gc-space (X; cl) if clD = X. Example 2.1 shows that in a gc-space (X; cl), an sgc-dense set may not be a gc-dense set. Note that if A = fag then clA = X but V \ A = ; for the nonempty open set V = fcg. In an isotonic space (X; cl), if A is an sgc-dense set then for any superset B of A, B is also an sgc-dense set. But if we consider example 2.1 then (X; cl) is not isotonic and if A = fag, B = fa; bg, we see clA = X, but clB 6= X. Theorem 2:1. If (X; cl) is isotonic then an sgc-dense set is gc-dense. Proof is easy. The converse of the above theorem may not stand. Example 2:2. Let X = fa; b; cg, cl; = ;, clfag = fa; bg, clfbg = fb; cg, clfcg = fb; cg, clfa; bg = X, clfa; cg = X, clfb; cg = fb; cg, clX = X. Then (X; cl) is isotonic and fag is gc-dense but not sgc-dense. Theorem 2:2. Let (X; cl) be a closure space. Then the following statements are equivalent for any subset A of X. (i) A is gc-dense (ii) A is sgc-dense. Proof. Clearly (ii) ) (i). For (i) ) (ii), let A be gc-dense but clA 6= X. Then X ¡ clA 6= ;) int(X ¡ A) 6= ;. Let V = int(X ¡ A) = X ¡ clA. Then V is open and nonempty, but V \ clA = ;. Since A ½ clA, then V \ A = ;, a contradiction to the fact that A is gc-dense. Thus (i) ) (ii). 184 Ch. Chattopadhyay Idempotent property and expanding property are not redundant in the above theorem. Consider Example 2.2. Let A = fag. Then cl(clA) = clfa; bg = X 6= clA. So (X; cl) is not idempotent but it is isotonic. We have already noted that A is gc-dense but not sgc-dense. Example 2:3. Let X = fa; b; cg. cl; = ;, clfag = fa; bg, clfbg = fcg, clfcg = fb; cg, clfa; bg = X, clfa; cg = X, clfb; cg = fb; cg, clX = X. Then (X; cl) is isotonic but not expanding since fbg is not a subset of clfbg. Note that fag is gc-dense which is not sgc-dense. In case of a topological space (X; ¿) it is true that for any nonempty open set V and for any nonempty subset A of X, V \ clA 6= ;, V \ A 6= ;. This is not true in case of gc-spaces. See example 2.1. Here fcg is open fcg \ clfag = fcg but fcg \ fag = ;. So it is quite natural to consider another type of dense set in a gc-space. Definition 2.3. In a gc-space (X; cl), a subset A of X is said to be a wgc- dense set if for every nonempty open set V in (X; cl), V \ clA 6= ;. In Example 2.1, since clfag = X, for every non-empty open set V , V \clfag 6= ;, but for the open set fcg, fcg \ fag = ;. Thus fag is a wgc-dense set but not gc-dense. In the same example, note that A = fa; cg is gc-dense but clA = fa; bg. Hence A is not wgc-dense because fcg is open and fcg \ clA = ;. In a gc-space, an sgc-dense set is necessarily wgc-dense. The converse may not be true. Consider Example 2.2. Here the non-empty open sets are fag and X. Consider A = fag. Then V \ clA 6= ; for every non-empty open set V , because clA = fa; bg. But clA 6= X. Theorem 2:3. Let (X; cl) be an idempotent gc-space. The following statements are equivalent for any non-empty subset A of X. (i) A is wgc-dense, (ii) A is sgc-dense. Proof is easy. We note that if a gc-space is expanding then the class of all gc-dense sets is contained in the class of all wgc-dense sets.
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